Positive $(p,n)$-intermediate scalar curvature and cobordism

Abstract

In this paper we consider a well-known construction due to Gromov and Lawson, Schoen and Yau, Gajer, and Walsh which allows for the extension of a metric of positive scalar curvature over the trace of a surgery in codimension at least $3$ to a metric of positive scalar curvature which is a product near the boundary. We generalize this construction to work for $(p,n)$-intermediate scalar curvature for $0\le p\le n-2$ for surgeries in codimension at least $p+3$. We then use it to generalize a well known theorem of Carr. Letting ${\mathcal{R}}^{s_{p,n}>0}(M)$ denote the space of positive $(p,n)$-intermediate scalar curvature metrics on an $n$-manifold $M$, we show for $0\le p\le 2n-3$ and $n\ge 2$, that for a closed, spin, $(4n-1)$-manifold $M$ admitting a metric of positive $(p,4n-1)$-intermediate scalar curvature, ${\mathcal{R}}^{s_{p,4n-1}>0}(M)$ has infinitely many path components.

References in corpus (11)

• Infinite loop spaces and positive scalar curvature
• The space of metrics of positive scalar curvature
• The Gromov-Lawson-Chernysh surgery theorem
• The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature
• Homotopy groups of the moduli space of metrics of positive scalar curvature
• The positive scalar curvature cobordism category
• Infinite loop spaces and positive scalar curvature in the presence of a fundamental group
• The action of the mapping class group on metrics of positive scalar curvature
• The Space of Positive Scalar Curvature Metrics on a Manifold with Boundary
• On the homotopy type of the space of metrics of positive scalar curvature
• On the space of riemannian metrics satisfying surgery stable curvature conditions